We analyse within first-order perturbation theory the instantaneous transition rate of an accelerated Unruh-DeWitt particle detector whose coupling to a massless scalar field on four-dimensional Minkowski space is regularised by a spatial profile. For the Lorentzian profile introduced by Schlicht, the zero size limit is computed explicitly and expressed as manifestly finite integral formula that no longer involves regulators or limits. The same transition rate is obtained for an arbitrary profile of compact support under a modified definition of spatial smearing. Consequences for the asymptotic behaviour of the transition rate are discussed. A number of stationary and nonstationary trajectories are analysed, recovering in particular the Planckian spectrum for uniform acceleration.

If you want to know what this is about without reading the paper, I wrote a couple of months ago a (rather) nontechnical explanation of my area of research (here), and a more technical summary of what my paper was to be about (here). Feedback from readers of the paper, and even suggestions for improvements (as it has not been sent for peer-review yet) are most welcome.

I'm enjoying reading your blog and I'm especially interested in what you ahve to say about the Unruh effect. Anyway, let me quote something I don't get:it implies that contrary to all physical intuition, the notion of particles is relative to the observer

I don't understand why you say this.

But the number of particles can't possibly be observer dependent, and nothing you have said suggests it is. An observer at rest looking at an accelerating detector travelling in a vacuum sees the detector detect particles with a thermal distribution. Accelerating a detector makes it 'detect'. An accelerating observer, and an observer at rest, will both agree that the detector has detected. So there is nothing observer dependent. Accelerated detectors behave differently from intertial detectors. This is completely consistent with SR and is an interesting non-trivial effect. It's not 'contrary to all physical intuition'. And there's no observer dependence in the number of particles detected. (Maybe you could argue over the number of undetected particles - but that's like arguing over how many angels are on a pinhead. What matters is what is detected.)

Thanks for your commenting; I have been a long-time lurker at your blog. As to your question, you are quite right in that in the particle detector approach, there is really nothing observer dependent in the concept of particles. This is because in this approach “particles” are defined operationally as nothing more than excitations in a detector, and the excitation is an objective fact. But another approach to the Unruh effect is to use the usual QFT non-operational definition of particles in terms of states in the Fock space; particles of a given momentum are states associated with field modes of a given wavelength. Here the Unruh effect arises because the Rindler vacuum associated with Rindler field modes (modes which are positive frequency with respect to the proper time of an accelerated observer) is inequivalent to the Minkowski vacuum associated with the usual plane wave modes (positive frequency with respect to the Minkowki time coordinate, the proper time of inertial observers). It turns out that when the field is in the Minkowski vacuum state, the number of “Rindler particles” present (calculated via a Bogolubov transformation) is nonzero, and thermal. It is commonly said that an accelerating detector responds to Rindler particles, though there are a lot of subtleties around this statement.

This is usually interpreted as proof that the “number of particles in the field” is observer-dependent quantity, much in the same sense as “time between two events” is an observer-dependent quantity in SR. Notice that for SR you can as well “go operational” about time, and talk only about how many clicks will there be a clock carried by each observer, and I will find the same answers for your clock than you do -answers different than those for my clock. So it seems to me there is about as much logic in saying that “number of particles depends on the acceleration of the observer” as in saying that “time between two events depends on the motion of the observer”, something we are quite happy to say in SR. If you are accelerated and I am at rest, you will see particles and I won't. Assuming your eyes work more or less like an Unruh detector, you may see directly the light of thermal photons. I can agree with you in the fact that you are seeing light; but it remains true that I don't see it myself.

You may find this paper to be of interest. It is mathematically sophisitcated and also contains much philosophical discussion:

The proper time measured measured by a clock moving along a path in spacetime depends on the path. Similarly, the number of particles detected by a detector in a vacuum depends on the path. But I'm not led to say that the density of particles in a vacuum is different when you're accelerating. If someone interprets the increased number of particles detected by an accelerating detector as the fact that somehow there are more particles in the vacuum when you accelerate, it's a bit like someone else saying they've reduced their mass when they decide to weigh themselves on a plane just as it decides to accelerate downwards. They simply measured incorrectly and need to calibrate their instruments taking into account the fact that they are accelerating.

Anyway, that paper looks great. I'll see how far I can get through it. And you can be sure I'll be coming back with questions some time soon.

I've always had a few problems with the Unruh effect, despite having read most of Wald's book on quantum fields in curved spacetime.

[Quote]"But I'm not led to say that the density of particles in a vacuum is different when you're accelerating."

As I see it,the key is that the "particles" both observers are talking about are different. The static one says "there are no Minkowski particles", the accelerated one says "there are Rindler particles".

[Quote]"If someone interprets the increased number of particles detected by an accelerating detector as the fact that somehow there are more particles in the vacuum when you accelerate, it's a bit like someone else saying they've reduced their mass when they decide to weigh themselves on a plane just as it decides to accelerate downwards. They simply measured incorrectly and need to calibrate their instruments taking into account the fact that they are accelerating."

This may sound sensible in flat spacetime, where there is a "privileged" set of observers, the inertial ones. We can feel it natural to decide that if inertial observers see no particles, then there are no "real" particles and what the accelerated observer sees is an illusion.

But what are we to do then in a general, curved spacetime? In a Schwarzchild black hole, for example, would you trust the measurements of an observer static with respect to the black hole (who sees thermal particles; this is the Hawking effect) even though this observer is not "inertial" but needs to accelerate constantly to avoid falling in? Or would you trust an observer freely falling into the black hole in geodesic movement? (The response of a particle detector in the latter motion is not known at present. A main motivation for my work is to find a way to calculate it!)

I think it's hard to answer this question:Or would you trust an observer freely falling into the black hole in geodesic movement?

until you've figured outThe response of a particle detector in the latter motion

But I'm inclined to say that if I were forced to define a notion of particle density that was independent of detector, then I would say that it would have to agree with what is detected by an inertial detector - ie. one freely falling into the black hole.

Sounds like you have a great research problem to work on - so good luck!

The question I was wondering about was this: stick with a flat spacetime for now. Suppose a detector moves along the path P(t). What does it detect? All of the papers I have seen discuss constant acceleration and these require integrals from -oo to +oo and so assume constant acceleration for all time. Is there some published work on this? Is it a reasonable approximation to say that at time t the detector will detect particles similarly to a constantly accelerating detector with the same instantaneous acceleration? If it is good, what is the domain in which this approximation is good?

Your question is actually an excellent one! There have been lots of studies of detectors with finite response time so the integrals don't go to infinity, but always moving on inertial or uniformly accelerated trajectories. To my knowledge the first study of general, nonstationary motion (trajectory not a Killing vector orbit) is Schlicht's

http://arxiv.org/abs/gr-qc/0306022

which had to develope a different regularization procedure to deal with this. My work is a generalisation and extension of his, and the formulas (6.1) and (6,2) in the conclusions of our paper summarise our results giving the response at a given time of detectors in arbitrary motion. They are in integral form, but some general properties of the result, as well as applications to concrete trajectories, are discussed in Section 5. Both numerical (Schlicht) and analytical (we) studies in detectors progressing from inertial to accelerated motion show a response growing gradually from zero to thermal, but I don't think the approximation you suggest works as a rule; the response depends on the whole past of the trajectory, not just the present acceleration.

The initial motivation that got me started on this problem was really the black hole situation, but it is much more difficult to work in Scwarzschild that in Minkowski. I have made some progress in de Sitter space as a first curved-space example, though. Thanks for the good luck wishes!

As to your intuition that the detector falling into the black hole is more trustworthy because it is inertial... well, on the other hand, that detector unlike the static one is not following a Killing vector orbit. So no strict definition of "particles" along that trajectory (beyond the purely operational one as number of clicks of the detector) is available in analogy with Minkowski or Rindler particles in flat space, which are defined by quantization of field modes positive-frecuency with respect to the time translation Killing vector and the boost Killing vector respectively.

These issues make problematic talk of "particles" as a fundamental notion in curved spaces, supporting in my opinion an interpretation of QFT with fields as ontologically real and particles defined operationally or only within approximations. But I think that insofar we keep talking of "particles" as real, the lesson of the Unruh effect is that they may be relative to an observer.

I think you've now given me enough reading for the long July 4 weekend coming up (in the US). Thanks!

I view the inability to define particles in most spacetimes as a proof that the concept of particle, apart from the operationalist one, is ill-defined. I really disliked the parts of Wald where he uses Killing vectors. It just seemed so wrong to me to have a definition of something that only works in spacetimes that have enough symmetry. But it's a couple of years since I read the book and I can't remember the exact details. If I get time to read these papers I'll be more competent to judge.

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Thoughts on physics, maths, science, philosophy, and anything else that may cross my mind, plus news about my current life for distant friends, by an Argentinian in the second third year of his PhD at the University of Nottingham.

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